where \(\mathcal {W}\)  denotes the Lambert-\(\mathcal {W}\)  function (also known as the ProductLog
function).
Although there exist a good number of solutions describing extended objects in theories of gravity
(see [1]}, [2]}, [3]}, [4]}, [5]} for some recent results in this area),
building solutions of this kind in four dimensions in not an easy task,
and usually requires the introduction of exotic matter fields [6]}, [7]}, [8]}.
Therefore, it is important to highlight that the black string constructed
here comes from the coupling between two very relevant physical theories, both from a theoretical and phenomenological point of view.
